CHAPTER 1
Basic Financial Concepts
1. Definitions
This is a mathematics course, in the sense that finance is used to learn
new mathematical concepts and methods. Before we start, we need to have
an appropriate vocabulary of financial terms. Each term has two aspects:
what it means in the real world and what it means mathematically.
Let’s start with some important terms commonly used in finance. Since
not all financial terms are formally defined in this book. It will be helpful to
have a dictionary of finance handy. Dictionary of Finance and Investment
Terms, by John Downes and Jordan Elliot Goodman, published by Barron’s,
2nd edition.
Stock. Ownership of a corporation represented by shares.
Option. A contract that gives the owner the right to buy or sell an
underlying asset at a fixed price K (strike price) before or at a fixed time T
(expiration). A put (option) is a right to sell, and a call is a right to buy.
American option. An option that can be exercised at any time prior to
expiration.
European option An option that can be exericised only at expiration.
Note that a holder of an option never loses money and has the potential
to make money. Therefore it is worth something, i.e., it has a price, which
is called the premium. If you purchase an option (instead of being given
one), the maximum loss you can have is the amount of the premium. The
premium is the value of an option (or of any financial instrument). It should
not be confused with commission , which is the fee you pay to a broker for
the service provided. When purchasing an option from a broker, you need
to pay both premium and commission.
To simplify the exposition, unless otherwise stated, we assume that the
market is frictionless, i.e., we ignore commissions in all our computations.
Another assumption we often use is that the market has infinite capacity,
i.e., you are able buy from the market or sell to the market any number of
shares of an underlying asset at the market price.
We will use the following notations for option prices:
European call option: c(t, T, K; S);
European put option: p(t, T, K; S);
American call option: C(t, T, K; S);
American put option: P (t, T, K; S);
1
2
1. BASIC FINANCIAL CONCEPTS
where the parameters are
t = current time,
T = expiration time,
K = strike price,
S = current asset price (spot price).
A parameter can be omitted if its value is understood. We will explain these
parameters in the next section.
Interest. The cost of using money or an asset, expressed as a rate per
period of time (usually a year). The nominal rate is the rate without compounding, and the effective rate is the rate after compounding.
Arbitrage. Profiting from differences in price of the same currency or
commodity traded on different markets. For example, if the price of gold is
higher in London than in New York one can buy in New York and sell in
London for a risk-free profit. That would be an arbitrage opportunity. While
these opportunities night exist, it would necessarily be of short duration.
Market observers attempting this strategy would drive up the price of gold
in New York with increased demand and lower the price of gold in London
with increased supply. This would end the arbitrage opportunity.
An arbitrage argument is an argument based on the principle that there
are no arbitrage opportunities without risk. Prices of many financial instruments are determined by an arbitrage argument. We will discuss this topic
later in great detail.
Bond. An interest-bearing corporate or government security with the interest and principal payable at a fixed time. It may also pay a cash dividend
(coupon) at regular time intervals during the life of the contract.
A bond has a face value (or par value), a maturity date, and a coupon
rate. For example, a bond with face value $5,000, maturity date August 1,
2021 and coupon rate 8% will allow the bearer to collect $200 (4%) every
6 months for the next 20 years and a maturity payment of $5,000 on the
maturity date. The current price of the bond may be $2,050. A zero-coupon
bound A bond that pays no coupons (cash dividend). Most of education
bonds are of this type.
Short sale. Sale of a security not owned by the seller in hopes of taking
advantage of an anticipated price decline. For the short seller, it is basically
a promise to deliver the security at a fixed price on a future date. If the
seller actually owns the security, his short position is said to be covered.
The concept of short sales is the stock equivalent of borrowing money from
a bank. Just as you can you borrow cash from a bank, we also assume that
you can borrow stocks from a stocker broker. But there is an important
difference. When you return a cash loan, you pay back the borrowed amount
(principal) and the interest for the loan period, whereas for a loan on a
stock, you return in kind, namely, you return the same number of shares
2. OPTION CONTRACTS
3
of the same stock to the lender. Of course, in practice you need to pay a
commission for the loan, but under our assumption of the frictionless market,
the commission is ignored.
Forward contract. An agreement to buy or sell a specific amount of a
commodity or financial instrument at the current price with delivery and
settlement at a future date.
Note that a forward contract is not an option, i.e., the transaction has
to take place at the specified time.
A large part of this course is devoted to option pricing theory developed
by Black, Scholes and Merton. The subject was born in 1973 with the papers “The pricing of options and corporate liabilities” in Journal of Political
Economics by Black and Scholes and “The theory of rational option pricing,” in Bell J. Econ. Management Sci. by Merton. In these papers, the
famous Black-Scholes formula for pricing options was derived. In the same
year, the Chicago Board Options Exchange (CBOE) opened. In the intervening years the Black-Scholes formula has become the accepted way to price
stock and other options. There are trillions of dollars involved in options exchanges yearly. Moreover, option pricing and risk management has become
a major area of applied mathematics. Research activity in Mathematical
Finance requires highly sophisticated tools from mathematics (mainly stochastic analysis) and is carried on in academia as well as in the financial
industry.
2. Option contracts
Here is a list of essential terms related to options, some of which you
are already familiar with from the last section. Consult a finance dictionary
and try to memorize their meanings. If you cannot find a term in your
dictionary, try to guess what it means. Dual terms are grouped together.
– Call option, put option
– Strike Price
– Expiration date (expiry)
– Exercise
– American option
– European option
– Option buyer, option writer
– In the money, at the money, out of the money
– Intrinsic value
– Premium
Intrinsic value is the profit to be made on an option at the time of
exercise. If the strike price of a call option is $50 and current market price
is $55, the intrinsic value is $5. If there is profit to be made, the option is in
the money; if exercising the option will incur a loss, the option is said to be
out of the money; if the strike price is equal to the market price, it is said
to be at the money .
4
1. BASIC FINANCIAL CONCEPTS
******
Reading assignment: Options for the Stock Investor, by James B. Bittman,
McGraw-Hill, New York (1997), Chapter 6.
******
What are options? We first recall the two types of options, a call option
and a put option. A holder of a call has the right to buy a specified asset at
a price determined by the option contract. A holder of a put has the right to
sell a specified asset at a price determined by the option contract. Options
can also be classified by the way they are exercised. For an American option,
the holder can exercise his right at any time up to time T . For a European
option, the holder can exercise his right only at time T . Other types of
options are called exotic options , including the Asian option, in which the
price to sell or buy the stock is the average of the stock price during the life
of the option instead of the market (or the spot) price.
The holder of an option contract is said to have a long position on the
option, meaning that he has purchased the option from an agent and has
paid an premium for it. In this case, the agent is said to hold a short position.
The asset specified in the option contract is called the underlying asset , or
the underlying for short.
The phrase “underlying asset” is a general and loose term to denote
the commodity specified in a contract. It usually conveys the idea that
the future price of this commodity cannot be predicted with certainty. But
mathematically it does not have to refer to something tangible. A contract
doesn’t have to be based on a tangible asset at all. For example, it can be
based on the S&P 500 index, or the temperature at noon on Wall Street on
June 1, 2005, or anything on which people have different opinion.
Two parameters in an options contract are its terminal or expiration
time T , sometimes also referred to as expiry or maturity, and the strike
price K, which is the agreed sale price in the option contract. To determine
the price of an option we need also need know at least the current time t
and the current stock price St .
Here are some very practical questions to consider at this point.
• Why would an agent enter an options contract?
• What does an agent have to gain from entering an options contract?
• What should an agent pay for an options contract?
There are two reasons agents enter into an options contract. The first
reason is that options can be used as a hedge against risk. These agents,
called hedgers, routinely face the problem of acquisition of goods, foreign
currency or loans. These goods may be necessary in the normal course of
their business activities. The fluctuations in price, exchange rate, interest
rate with time are random, or at the very least unpredictable. An options
contract can act as insurance against unfavorable price fluctuations.
A second reason for entering an options contract is to gamble on the
price fluctuation of stocks for possible significant profits. For this the agent
3. EXAMPLES OF OPTIONS
5
(a speculator) has to be willing to accept risk. A hedger in effect transfers
(or sells) his risk to a speculator. Hedging by purchase of an options contract
is tantamount to buying insurance, which is not free. The advantage of a
call option over outright purchase of the stock is that with a call option the
loss is capped at the premium of the option (which would be the loss in the
long position of an unexercised option), whereas with the stock the loss can
be the whole purchase price of the stock.
Let’s assume that we have purchased a European call option at time
t = 0. We denote its value at time t by c(t, T, K, St ). Thus c(0, T, K, S0 )
is the premium paid at t = 0 for the option. The profit from this option
at the expiration is easy to write down. First we introduce two functions.
We use x+ to denote 0 if x < 0 and x otherwise. This function is called the
positive part of x. The negative part of x is the function x− , which has the
value 0 if x > 0 and −x otherwise. Note that these two functions satisfy the
relations.
x = x+ − x− ,
|x| = x+ + x− .
At time T , if the option is in the money, meaning ST ≥ K, we exercise the
option and have the profit ST − K; otherwise we let the option expire and
the profit in this case is 0. It follows that the value of the option at time T
is just (ST − K)+ . This can be written as
c(T, T, K, ST ) = (ST − K)+ .
A similar argument can be applied to a put option:
p(T, T, K, ST ) = (K − ST )+ .
Of course, the cost of purchasing the contract (premium plus commission)
must be subtracted to obtain the net profit.
We have seen that the function c(t, T, K, S) has a terminal value
c(T, T, K, S) = (S − K)+ .
What we are interested in is its initial value c(0, T, K, S). After all we need to
determine the fair price for these instruments. One of our goals in this course
is to develop a theory which enable us to determine this function completely.
But even without knowing this function explicitly, we can use some common
sense (and some not so common sense) to derive several important properties
this function must satisfy. This will be done in the next few sections.
3. Examples of options
The value of a call option c(t, T, K, S) is a decreasing function of K,
because the option is less valuable if you have to pay a higher price when
you exercise it. Symbolically,
c(t, T, K1 , S) ≥ c(t, T, K2 , S)
if
K1 ≤ K2 .
6
1. BASIC FINANCIAL CONCEPTS
Example 3.1. (Bull spread) Suppose you are bullish on a given stock,
that is, you expect that its value will rise. Can you use options to profit
from a rise in the stock’s price? Here’s what you might try:
(a) Buy a call with strike price K1 .
(b) Sell a call with strike price K2 > K1 .
Suppose that the value of the call options are c(Ki ) = c(t, T, Ki ), i = 1, 2.
We have just seen that
c(K2 ) ≤ c(K1 ).
Let P (t) denotes the value of this portfolio at time t. Then
P (t) = c(t, T, K1 , St ) − c(t, T, K2 , St ).
At t = 0,
P (0) = c(K1 ) − c(K2 ) ≥ 0,
which is the cost to establish the portfolio. At expiration
P (T ) = (ST − K1 )+ − (ST − K2 )+ .
This function is nonnegative function of the stock price ST (see the diagram).
[Diagram of f (s) = (s − K1 )+ − (s − K2 )+ .]
The maximum loss of the bull spread is limited to c(K1 ) − c(K2 ) if the stock
price decreases below K1 , and the maximum profit is limited to K2 − K1 −
c(K1 ) − c(K2 ) if the stock rises above K2 .
Compare the bull spread with a straightforward call option, which can
also be used to profit from a rise in the stock’s price.
Example 3.2. (Bear spread) On the other hand if you’re pessimistic
about the outlook for a stock, you can construct a portfolio of options to
profit in case you turn out to be correct. Try reversing the procedure with
the bull spread. That is
(a) Buy a put with strike price K2 .
(b) Sell a put with strike price K1 < K2 .
The payoff of the option is
P (T ) = (K2 − ST )+ − (K1 − ST )+ .
[Diagram of f (s) = (K2 − s)+ − (K1 − s)+ .]
The cost of establishing this portofolio is
P (0) = p(0, T, K2 , S0 ) − p(0, T, K1 , S0 ).
Since the payoff is nonnegative we have
p(0, T, K2 , S0 ) ≥ p(0, T, K1 , S0 ),
Namely, the value of a put option is an increasing function of the strike
price.
3. EXAMPLES OF OPTIONS
7
Example 3.3. (Butterfly spread) Let’s now discuss the so-called butterfly spread . This time the agent expects the underlying asset price will
not move very much between the times 0 and T . He does the following:
a. Buy a call with strike K1 .
b. Buy a call with strike K2 > K1 .
c. sell two calls with strike K3 = (K1 + K2 )/2.
In this case, the payoff at expiration is
K1 + K2 +
+
+
.
P (T ) = (ST − K2 ) + (ST − K1 ) − 2 ST −
2
The cost of establishing this portfolio is
C = c(K1 ) + c(K2 ) − 2c(K3 ).
The diagram of the payoff as a function of the stock price ST is
[Diagram of f (s) = (s − K2 )+ + (s − K1 )+ − 2 s −
K1 +K2 +
.]
2
Note that the payoff function is nonnegative, i.e., this portfolio can never lose
money. It follows that the cost of establishing this portfolio is nonnegative,
namely, C ≥ 0 in this case, or
K1 + K2
c(K1 ) + c(K2 )
.
c
≤
2
2
A function with this property is called a convex function.
A function f (x) on an interval [a, b] is called convex if for any a < x, y <
b, the graph of the function on the interval [c, d] lies below the straight line
joining the two points (x, f (x)) and (y, f (y)) on the curve. This condition
can be written as follows: for any x, y ∈ [a, b] and any 0 ≤ λ ≤ 1,
f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y).
When λ = 1/2, we have
f
x+y
2
≤
f (x) + f (y)
.
2
In fact, it can be shown that this special case implies that the above inequality holds for all 0 ≤ λ ≤ 1. Hence we have the following conclusion.
Theorem 3.4. The price of a call option c(K) = c(0, T, K, S) is a decreasing, convex function of the strike price K.
For a smooth function (with continuous second derivative), there is a
simple condition for convexity.
Theorem 3.5. If the second derivative f 00 is nonnegative, then f is
convex.
If −f is a convex function, then f is called a concave function.
8
1. BASIC FINANCIAL CONCEPTS
Example 3.6. (Straddle) Now the agent expects a big change in the
stock price, though he is uncertain in which direction. This time the agent
establishes the following portfolio:
(a) Buy a call with strike K.
(b) Buy a put with strike K.
Then his payoff at expiration is
P (T ) = (ST − K)+ + (K − ST )− = |ST − K|.
[Diagram of f (s) = |s − K|.]
In this case the holder of the above portfolio stands to gain when the stock
price makes either an up or down movement.
Example 3.7. (Strangle) The agent has the feeling that the asset price
will move well out of its present range. Accordingly he creates a portfolio
composed of
(a) A long position in a call with strike K2 .
(b) A long position in a put with strike K1 < K2 .
Here K1 < S0 < K2 , where S0 is the stock price at time t = 0. This
combination resembles a straddle, but the diagram between K1 and K2 is
flat. The time T payoff is
P (T ) = (ST − K2 )+ + (K1 − ST )+
[Diagram of f (s) = (s − K2 )+ + (K1 − s)+ .]
The payoff for a straddle with K = (K1 + K2 )/2 always exceeds the payoff
for a strangle with strikes K1 and K2 . The price of a straddle must therefore
be greater than the price of a strangle.
Spreads are strategies of buying and selling the same options at different strike prices and/or with different expiration times. They are further
classified as follows:
vertical spread: different strike prices and same expiration time;
horizontal spread: same strike price and different expiration times;
diagonal spread: different strike prices and different expiration times.
Horizontal and diagonal spreads are collectively referred to as calendar
spreads. Here is an example of calendar spreads.
Example 3.8. (Calendar spread) Our agent now wants to profit in case
the asset price is low at T1 and rises at a later time T2 . This calls for the
following portfolio:
(a) Sell a call with strike K and expiration T1 .
(b) Buy a call with strike K and expiration T2 .
The resulting payoff is (ST2 − K)+ − (ST1 − K)+ . It depends on the asset
prices at both time T1 and T2 . If the graph of St is below K at T1 and above
K at T2 , the agent will receive ST2 − K.
4. EXERCISES
9
4. Exercises
Exercise 4.1. John Doe sold short 3,000 shares of XYZ stock short at
$72 per share. A month later, a false alarm caused the stock price to plunge
to $45 per share, at which John Doe bought 3,000 shares to cover his short
position. He then bought 3,500 shares of the same stock at $52, and later
sold these shares at $113 when the price quickly rebounded. How much
profit did John Doe make on XYZ?
Exercise 4.2. Follow the proof for the European call option price function to show that the European put option price p(K) = p(t, T, K, S) is an
increasing, convex function of K.
Exercise 4.3. (Strap) An agent is fairly confident that a stock price
will move, but wishes to profit more from an upturn, which he feels is more
likely than a downturn. He acquires the following portfolio:
(a) Buy two calls with strike K and maturity T .
(b) Buy one put with maturity K and maturity T
[Graph of the function f (s) = 2(s − K)+ + (K − s)+ .]
This gives a gave similar to that for the straddle, but the slope of the graph
to the right of K for the strip is twice as large as for the straddle. If the
strike price at expiration is $120, what is the payoff of a strap with strike
price $130?
Exercise 4.4. (Strip) An agent has similar expectations as for the strip
except he is more confident that a downturn will occur than an upturn. This
agent acquires the portfolio
(a) Buy a call with strike K and expiration T .
(b) Buy two puts with strike K and expiration T .
The graph of the payoff for this combination is obtained from the graph
of the strip by reflecting in the vertical line s = K. If the stock price at
expiration is $50, what is the payoff with a strike price $35?
Exercise 4.5. Let K1 < K2 < K3 . Show that the three call option
prices c(Ki ) = c(0, T, K, S) satisfies
K3 − K2
K2 − K1
· c(K1 ) +
· c(K3 ).
c(K2 ) ≤
K3 − K1
K3 − K1
If we let λ = (K3 − K2 )/(K3 − K1 ), then 0 ≤ λ ≤ 1 and
c(λK1 + (1 − λ)K3 ) ≤ λc(K1 ) + (1 − λ)c(K3 ),
which shows directly that c(K) is a convex function of K.